A synthesis filter bank
corresponding
to analysis filter bank
is defined as that filter bank
which inverts the analysis filter bank, i.e., satisfies

Clearly, not every filter bank will be invertible in this way. When
it is, it may be called a perfect reconstruction filter bank. When
a filter bank transfer function
is paraunitary, its
corresponding synthesis filter bank is simply the paraconjugate filter
bank
, or

The channel filters
in a paraunitary filter bank
are power complementary:

This follows immediately from looking at the paraunitary property on the
unit circle.

When
is FIR, the corresponding synthesis filter
matrix
is also FIR. Note that this implies an FIR
filter-matrix can be inverted by another FIR filter-matrix. This is in
stark contrast to the case of single-input, single-output FIR filters,
which must be inverted by IIR filters, in general.

When
is FIR, each synthesis filter,
, is simply the
of its corresponding
analysis filter
:

where
is the filter length. (When the filter coefficients are
complex,
includes a complex conjugation as well.)

This follows from the fact that paraconjugating an FIR filter amounts
to simply flipping (and conjugating) its coefficients.

Note that only trivial FIR filters can be paraunitary in the
single-input, single-output (SISO) case. In the MIMO case, on the
other hand, paraunitary systems can be composed of FIR filters of any
order.

FIR analysis and synthesis filters in paraunitary filter banks
have the same amplitude response.